Types of Cones

Date: 01/19/99 at 15:57:04
From: Jamie Rowan
Subject: Edges of cones
Does a cone have an edge?

Date: 01/20/99 at 08:55:54
From: Doctor Rob
Subject: Re: Edges of cones
There are several meanings to the word "cone." In some, it does not
have an edge; in others, it does. I will try to describe the various
meanings precisely and perhaps then you will see what I mean.
Set up a Cartesian rectangular xyz-coordinate system as follows. Take
the origin to be the vertex of the cone, and the z-axis to be the axis
of the cone. Then a right circular cone can be defined by an equation
of the form
a^2*z^2 = x^2 + y^2
for some nonzero constant a. The words "right circular" mean that a
cross-section perpendicular (at a right angle) to the axis is a circle.
Often these words are omitted, since it is quite uncommon to see an
oblique circular cone or a right elliptical cone, or other types. An
"infinite right circular cone of two sheets" consists of all points
whose coordinates satisfy this equation. In this form, you can see that
there is no limit on the size of z, so it extends infinitely in both
positive and negative z-directions; hence the word "infinite." The
words "of two sheets" mean that if you remove the vertex, the surface
is split into two connected parts disjoint from each other.
If you restrict z to z >= 0, then you get an "infinite right circular
cone of one sheet."
If you restrict z to 0 <= z <= b, for some positive b, then you get
part of a right circular cone of one sheet. I would call this a "finite
right circular cone of one sheet."
If you add the set of points z = b, x^2 + y^2 <= a^2*b^2 (the base of
the cone), you get a closed surface having two faces, with an interior
region. I would call this a "closed right circular cone of one sheet."
If you consider the region of xyz-space enclosed by that surface, that
is, points whose coordinates satisfy 0 <= z <= b, x^2 + y^2 <= a^2*z^2,
you get a "right circular conical region," or "solid right circular
cone."
Any of the above can be loosely called a "cone." Only the last two
have an edge, the circle z = b, x^2 + y^2 = a^2*b^2, with center
(0,0,b) and radius a*b.
The moral of this is that you may need to be precise about which of the
above "cones" you mean, then the answer as to whether or not there is
an edge can be easily determined.
- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/